Studying the multiscale nature of systems in science and engineering has led to many suc-cess stories, both in theory and application. The world seems inherently multiscaleat least to understand complex systems holistically, elements have to be grouped together on larger and larger scales. What is a scale? An informal denition is possible through consideration of spatial and temporal scales. A spatial scale is best described through the measurement tools needed to capture it adequately, namely the size of rulers, mea-suring rodsor scales used to measure distances in the given space, on the given scale.
Measuring the width of an atom with a ruler for architectural drawings does not make sense, neither does looking at geography with the precision of an electron microscope.
The same holds for temporal scales: switching parts of cellular molecules often occurs on temporal scales far below microseconds and needs very precise instruments to observe, whereas the current day of the year could already be measured with instruments thou-sands of years ago. All dynamic processes in nature happen on one or more temporal and spatial scales, and scientic descriptions, called models, must incorporate this.
Models are the basic tools to foster understanding of systems in the natural sciences.
A model is an abstraction of reality such that key features of the system at hand can be reproduced well, whereas others are neglected to simplify the model's description (Frigg and Hartmann, 2012; Bungartz et al., 2014). When it comes to capturing a system with a hierarchy of interacting scales, models with the same structure are often found to be superior to models with only one scale. The work of Shalizi (2006) provides a broad overview about the methods and techniques of complex systems science, including multiscale systems.
If a multiscale model has two scales, the scales are often called microscopic (or fast) and macroscopic (or slow). Models with scales in between the two are sometimes called mesoscopic models. We choose the ow of water particles in a river as an example. This system could be captured by a purely microscopic model of individual water molecules, which interact with their neighbors and are subject to gravity. The same system can be described macroscopically, through the ow of a certain volume of water over time. A multiscale model would take into account both scales: the water molecules are modeled explicitly and give information about the local ow to the macroscopic scale, where it is integrated into the global ow of volume. Li et al. (2004) review multiscale methodology for complex systems, and E (2011) provides an overview of numerical multiscale methods.
There is another challenge to multiscale modeling: given a description of a system on a certain scale, how to nd a description on another, coarser scale? This challenge is called homogenization (Stuart and Humphries, 1996; Givon, Kupferman, and Stuart, 2004) or upscaling (Farmer, 2002; Brandt, 2005), and is the main question addressed and answered in this thesis.
In the example of owing water, the challenge is to bridge spatial and temporal scales of many magnitudes. Models of continuous ow were formed successfully by upscaling systems analytically, usually in the limit of an innite number of particles. This was in part possible by the particles being simple molecules, with the same form and behavior.
This allows for strong assumptions on averages, which ultimately yield the macroscopic dynamic (Legoll and Lelièvre, 2010).
Generally, the more complex the individual particle, the more complex the behavior of a system with more of these particles. The overall number of particles in the system can also matter greatly when considering homogenization. In granular ow, the sizes of individual particles can vary greatly, and the number of particles drops from approxi-mately 1×1022 molecules in a gram of water to only a few millions or even thousands in a silo lled with grains. In this case, the assumption of equality of particles seizes to hold, and averaging produces results dierent from the actual process. The complexity of individuals can also increase when they are self-propelled, see the work of Helbing (2001) and Carrillo, Martin, and Panferov (2013) for reviews.
In car trac, the individual particles are human drivers in their vehicles (Bellomo and Dogbe, 2011). They are very complex individually, but generally all abide the rules of trac. Additionally, a car has very limited degrees of freedom when driven properly on a road. Cars are quite similar in speed and size compared to the size dierences of rocks in avalanches, and could hence be studied using tools from granular ow. The complexity of human drivers, and the fact that cars are self-propelled and not force-driven particles, makes car trac an active area of research on its own.
In crowd dynamics, the number of people moving is often comparable to the number of cars on a highway. However, pedestrians have more freedom to move than cars on roads, and change their direction of movement much quicker. At the same time, chal-lenges introduced by individual complexity remain. Thus, the scale transition from micro to macro in crowd dynamics is even more challenging than for car trac. Another major challenge, not only for upscaling, is the lack of a general purpose, microscopic model. This
is due to the intrinsic complexity of humans, with current models incorporating not only physical but also psycho-social eects. Comparing recent microscopic and macroscopic models, Duives, Daamen, and Hoogendoorn (2013) even state that for practical applica-tions, that need both precision and speed, the current pedestrian simulation models are inadequate. The multiscale nature of crowds has already been recognized in psychol-ogy, with the advance of theories such as social identity (Turner et al., 1987; Reicher, Spears, and Haslam, 2010). In short, and restricted to a crowd that is present physically, the theory explains crowd behavior macroscopically through the formation of dierent social groups. Individuals identify with one of the social groups, and act according to the norms of this group. The introduction of the social groups in addition to individuals makes social identity theory eectively a multiscale model of crowds. The complexity of humans results in a large number of parameters necessary to describe individuals.
The number of parameters must then be multiplied thousands of times for large crowds.
Most modeling attempts overcome this by using the same parameters for all individuals involved. This is in sharp contrast to homogeneous, molecular particle systems, where the repulsion and attraction potentials can be set with only a few parameters, and are the same for all particles. Due to the large number of challenges in crowd dynamics, many scientic elds are involved in its research: from mathematics (Francescoa et al., 2011; Degond et al., 2013), physics (Helbing and Molnár, 1995; Karamouzas, Skinner, and Guy, 2014), biology (Smith et al., 2007; Moussaïd et al., 2012), computer science (Richmond and Romano, 2008; Sud et al., 2008), engineering and safety science (Smith et al., 2009; Sivers et al., 2016), to psychology and sociology (Sime, 1995; Drury and Reicher, 2010).
Generally, observations of complex systems, followed by modeling, simulation, and analysis, can lead to predictions performed by computers (Sacks et al., 1989). Computa-tional challenges in the context of crowd dynamics are (faster than) real time simulations (Richmond and Romano, 2008; Mroz, Was, and Topa, 2014), and uncertainty quanti-cation (Iaccarino, 2008; Smith, 2014). The concept of data-driven surrogate models developed in this thesis helps to resolve the issue of real time simulations of macroscopic data, and also enable real time uncertainty quantication.
Finding the macroscopic equations of a system in closed form would ease the compu-tational burden, and there are already great successes for classic physical systems with a large number of particles. Data-driven upscaling can help pave the way to understanding, and numerical algorithms already incorporate multiscale ideas very successfully.
The thesis combines results from manifold learning and the theory on dynamical systems with state-of-the-art microscopic models in granular systems, such as crowd dynamics, car trac, and a generic granular ow model. The combination is a surrogate model approach to upscaling, where the microscopic model generates the data needed to learn the macroscopic model, eectively performing a scale transition through focusing on the observables that change the slowest.